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Preface to the English Edition I completed myself, and the translation of the" Introduction" and of this preface was done under the supervision of Springer-Verlag.I thank all who have worked on this edition, especially the editorial and production staff of Springer-Verlag. July 1984 o. A. LADYZHENSKAYA Contents Contents CHAPTER III Equations of Parabolic Type §1. Posing Initial-Boundary Value Problems and the Cauchy Problem §2. First Initial-Boundary Value Problem for the Heat Equation §3. First Initial-Boundary Value Problem for General Parabolic Equations §4. Other Boundary Value Problems. The Method of Fourier and Laplace. The Second Fundamental Inequality §5. The Method of Rothe Supplements and Problems CHAPTER IV Equations of Hyperbolic Type §l. General Considerations. Posing the Fundamental Problems §2. The Energy Inequality. Finiteness of the Speed of Propagation of Perturbations. Uniqueness Theorem for Solutions in w~ §3. The First Initial-Boundary Value Problem. Solvability in Wi(QT) §4. On the Smoothness of Generalized Solutions §5. Other Initial-Boundary Value Problems §6. The Functional Method of Solving Initial-Boundary Value Problems §7. The Methods of Fourier and Laplace Supplements and Problems CHAPTER V Some Generalizations §l. Elliptic Equations of Arbitrary Order. Strongly Elliptic Systems §2. Strongly Parabolic and Strongly Hyperbolic Systems §3. Schrodinger-Type Equations and Related Equations §4. Diffraction Problems Supplements and Problems CHAPTER VIThe Method of Finite Differences §1. General Description of the Method.

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We prove a criterion of local Hölder continuity for suitable weak solutions to the Navier-Stokes equations. One of the main part of the proof, based on a blow-up procedure, has quite general nature and can be applied to other problems in spaces of solenoidal vector fields. Mathematics Subject Classification (1991). 35K, 76D.In the present paper we deal with weak solutions to the three-dimensional NavierStokes equationsHopf proved the global existence at least one weak solution v to the first initial boundary value problem with boundary condition v| ∂Ω×[0,T ] = 0 under quite general assumptions on domain Ω, external force f and initial data v| t=0 = a. In [5] his results were discribed and the class of Hopf's solutions was introduced. Corresponding definition includes all main properties that essentially were proved by E. Hopf. More precisely, a velocity field v is called Hopf's solution if it belongs to T ] in the weak topology of L 2 (Ω; R 3 ) and satisfies the integral identityfor all t ∈ [0, T ] and for all w ∈ • J 1 2 (Ω). No information on the pressure p is given. Here • J(Ω) is the L 2 (Ω; R 3 )-closure of the set of all smooth solenoidal fields

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